lecture notes
The actual notes for this course begin with Chapter 56, on page 493.
The notes were last updated on 27.06.2025 and now contain everything up to the lecture
of 26.06.2025 (stable homotopy groups and spectra).
You may want to press the reload button
to make sure you are seeing the current version.
What appears before Chapter 56 is a complete set of notes that
I wrote for Topology I-II in previous semesters. This is
convenient to have there so that I can occasionally refer to it in the current course,
but that does not mean that you must already know most of what is contained in the
earlier parts of the notes. (See Prerequisites below for more on that.)
Problem Sets
Problem sets are generally posted here by the end of each week and are to be discussed in the
Übung the following week. Exercises in the lecture notes that are marked with an asterisk
should be considered essential; they typically concern proofs of results that will be
used later in the course.
- Problem Set 1 (to be discussed 24.04.2025): All exercises in Chapters 56 and 57 of the lecture notes.
- Problem Set 2 (to be discussed 15.05.2025): All exercises in Chapters 58, 59, 60 and 61 of the lecture notes.
Note: This set is a special case, because we had two Thursdays in a row that were holidays. Since it would
be impossible to cover the problems from four lectures in one problem session, written solutions to select
problems have now been posted on the moodle.
- Problem Set 3 (to be discussed 22.05.2025): All exercises in Chapters 62 and 63 of the lecture notes.
- Problem Set 4 (to be discussed 5.06.2025): All exercises in Chapters 64, 65 and 66 of the
lecture notes
- Problem Set 5 (to be discussed 12.06.2025): All exercises in Chapters 67 and 68 of the
lecture notes
- Problem Set 6 (to be discussed 19.06.2025): All exercises in Chapters 69 and 70 of the
lecture notes
- Problem Set 7 (to be discussed 26.06.2025): All exercises in Chapters 71 and 72 of the
lecture notes
Note: Exercise 71.3 used to appear in Chapter 69 and was already discussed in the session on 19.06.
I've displaced it now because I'd forgotten that its solution uses the simply connected case of the
relative Hurewicz theorem.
- Problem Set 8 (to be discussed 3.07.2025): All exercises in Chapters 73 and 74 of the
lecture notes
For general information about the course, scroll down a bit...
Announcements
- 25.04.2025: A major adjustment has been made to the schedule for the Übung,
and also minor adjustments for the lectures; see the red text below
under "Time and place".
- 13.04.2025: The problem class (Übung) will not meet in the first
week of the semester.
|

image by Christian Lawson-Perfect from
cp's mathem-o-blog
|
General information
Instructors:
- Prof. Chris Wendl (lectures); for contact information and office hours see my homepage
- Naageswaran Manikandan (problem class)
Moodle:
https://moodle.hu-berlin.de/course/view.php?id=133147
The enrollment key is: obstruction
Important: You must join the moodle for the course in order to receive occasional
time-sensitive announcements, e.g. if a lecture has been cancelled or rescheduled.
HU students can access moodle using their HU username and password.
Non-HU users can access it by following this link
and then clicking on "Create new account".
You will need to enter the enrollment key printed above.
Time and place:
Lectures on Wednesdays and Thursdays 11:00-12:30 in room 1.012 (RUD25)
Problem Class (Übung) Thursdays 13:30-15:00 in room 1.012 (RUD25),
starting in the second week of the semester
Note: Some minor adjustments have been made to the schedule since the start
of the semester, and are indicated above in red. Note in particular that both lectures
start on the hour, not 15 minutes after.
Language:
The course will be taught in English.
Prerequisites:
The main prerequisites are a solid foundation in point-set topology,
the fundamental group and covering spaces, singular and cellular (co-)homology
(including computations based on the axioms and some homological algebra,
e.g. the universal coefficient theorems), and some willingness to put up with the
language of categories, functors and universal properties. If you have taken my
Topologie II course before,
then you definitely have the essential prerequisites, but you probably also have
them if you learned about homology and cohomology elsewhere.
Some knowledge of smooth manifolds will occasionally be useful, but if you do not
have this, you will just need to be willing to accept a small set of facts about
tangent spaces, tubular neighborhoods and transversality as black boxes.
Contents:
This is a course on intermediate-level algebraic topology, and
is conceived in part as a sequel to
last semester's Topology 2 course.
We aim to prove some useful results from
elementary homotopy theory (some of which were mentioned briefly last semester
but were not proved), and introduce the essentials of obstruction theory,
classifying spaces, characteristic classes, and bordism theory.
These are all topics that have wide-ranging applications to other areas
of mathematics, especially to problems in differential geometry and topology,
such as the existence and classification of exotic smooth structures
on manifolds. We will not attempt any deep exploration of modern
homotopy theory, as that would far exceed my expertise.
Here is a more detailed plan of topics, though I cannot promise that all of them will
be covered.
- Notions from category theory: limits and colimits,
(co-)products, fiber products / pullbacks and pushouts, (co-)equalizers
- Point-set topological subtleties: evaluation maps, adjoint functors, quotient maps,
the compactly-generated category
- Elementary homotopy theory: mapping cylinders and cones, reduced suspensions, loop spaces
and adjunction, fibrations and cofibrations, homotopy (co)fibers,
transport functors, the Puppe sequences, group and cogroup objects
- Higher homotopy groups: definitions and group structure, exact sequences,
Serre fibrations, homotopy excision theorem, Freudenthal suspension theorem
- Homotopy theory of CW-complexes: weak homotopy equivalences and
Whitehead's theorem, n-connectedness, cellular and CW-approximation,
Eilenberg-MacLane spaces, homotopical characterization of cellular cohomology,
Hurewicz theorem
- Generalized (co)homology theories: (co)homology for pointed spaces, stable homotopy groups,
spectra
- Topology of fiber bundles: vector/principal/fiber bundles and structure groups,
induced bundles and homotopy invariance, universal bundles and classifying spaces,
reduction of structure groups, stable equivalence and K-groups
- Obstruction theory: singular and cellular (co-)homology with local coefficients,
obstruction cocycles for lifting/extension problems
- Characteristic classes: Chern, Stiefel-Whitney, Pontryagin and Euler classes,
the splitting principle, basic computations
- Bordism theory: definitions as generalized homology,
stable Pontryagin-Thom theorem,
tangential structures, computation of the rational oriented bordism ring,
Hirzebruch signature theorem
Literature:
The top of this page contains a link to detailed lecture notes for this
course which will be updated routinely as the course progresses.
The bulk of what we plan to cover is in any case contained in the union of the
following three books:
Each of these books has its own set of advantages and disadvantages, and none
contains absolutely everything that I'm hoping to cover; each also contains plenty of interesting stuff
that we will not have time for. Here are some other sources that are good to know about:
Here are some references that we are less likely to make much direct use of, but it would be wise to
remember that they are also out there:
Homework:
Weekly problem sets will be posted near the top of this page, and some
subset of those problems will be discussed in the problem class
(Übung) each week. Homework will not be collected or graded.
The problem class may sometimes also be used
to fill in gaps on details that did not fit into the regular lectures.
Grades:
Since this is an advanced course, I have a fairly relaxed attitude about
grades. If you come to the course with adequate prerequisites and stay with
it for the whole semester, you can come to my office at the end
for a conversation (let's pretend that's the English translation of
“mündliche Prüfung”). The format is as follows:
you pick one particular coherent topic from the course to focus on, typically the contents
of four to six lectures, and we will talk about that.
If you demonstrate that you learned something
interesting from the course, you'll get a good grade.
Chris Wendl's homepage